New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

References

• Apéry, R. (1979). Irrationalité de ζ(2) et ζ(3). Astérisque, 61:11–13.
   Google Scholar
• Apéry, R. (1981). Interpolation de fractions continues et irrationalité de certaines constantes. In Mathematics, CTHS: Bull. Sec. Sci., III. Paris: Bib. Nat.; pp. 37–53.
   Google Scholar
• Amdeberhan, T., Tauraso, R. (2016). Supercongruences for the Almkvist-Zudilin numbers. Acta Arith. 173(3):255–268.
   Web of Science ®Google Scholar
• Almkvist, G., van Straten, D., Zudilin, W. Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations. Proc. Edinb. Math. Soc. (2), 54(2):273–295. doi:10.1017/S0013091509000959
   Google Scholar
• Almkvist, G., Zudilin, W. (2006). Differential equations, mirror maps and zeta values. In Mirror Symmetry. V, Vol. 38 of AMS/IP Stud. Adv. Math. Providence, RI: Amer. Math. Soc.; pp. 481–515.
   Google Scholar
• Bostan, A., Boukraa, S., Christol, G., Hassani, S., Maillard, J.-M. (2013). Ising n-fold integrals as diagonals of rational functions and integrality of series expansions. J. Phys. A. 46(18):185202, 44.
   Google Scholar
• Bostan, A., Boukraa, S., Maillard, J.-M., Weil, J.-A. (2015).Diagonals of rational functions and selected differential Galois groups. J. Phys. A 48(50):504001, 29.
   Google Scholar
• Bostan, A., Chyzak, F., van Hoeij, M., Pech, L. (2011/12). Explicit formula for the generating series of diagonal 3D rook paths. Sém. Lothar. Combin. 66:Art. B66a, 27.
   Google Scholar
• Beukers, F. (1987). Another congruence for the Apéry numbers. J. Number Theory 25(2):201–210. doi:10.1016/0022-314X(87)90025-4
   Web of Science ®Google Scholar
• Beukers, F. (1987). Irrationality proofs using modular forms. Astérisque (147/148):271–283, 345. Journées arithmétiques de Besançon (Besançon, 1985).
   Web of Science ®Google Scholar
• Bostan, A., Lairez, P., Salvy, B. (2017).Multiple binomial sums. J. Symbolic Comput. 80(part 2): 351–386. doi:10.1016/j.jsc.2016.04.002
   Google Scholar
• Borwein, J. M., Nuyens, D., Straub, A., Wan, J. (2011). Some arithmetic properties of short random walk integrals. Ramanujan J. 26(1):109–132. doi:10.1007/s11139-011-9325-y
   Web of Science ®Google Scholar
• Viggo Brun, J., Stubban, O., Fjeldstad, J. E., Tambs Lyche, R., Aubert, K. E., Ljunggren, W., Jacobsthal, E. (1952). On the divisibility of the difference between two binomial coefficients. In Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949. Johan Grundt Tanums Forlag, Oslo; pp. 42–54.
   Google Scholar
• Borwein, J. M., Straub, A., Vignat, C. (2016). Densities of short uniform random walks in higher dimensions. J. Math. Anal. Appl., 437(1):668–707.
   Web of Science ®Google Scholar
• Borwein, J. M., Straub, A., Wan, J. (2013).Three-step and four-step random walk integrals. Exp. Math. 22(1):1–14.
   Web of Science ®Google Scholar
• Chowla, S., Cowles, J., Cowles, M. (1980). Congruence properties of Apéry numbers. J. Number Theory 12(2):188–190. doi:10.1016/0022-314X(80)90051-7
   Web of Science ®Google Scholar
• Chan, H. H., Cooper, S., Sica, F. (2010). Congruences satisfied by Apéry-like numbers. Int. J. Number Theory 6(1):89–97. doi:10.1142/S1793042110002879
   Web of Science ®Google Scholar
• Christol, G. (1984). Diagonales de fractions rationnelles et equations différentielles. In Amice, Y., Christol, G., Robba, P., eds. Study group on ultrametric analysis, 10th year: 1982/83, No. 2, pages Exp. No. 18, 10. Paris: Inst. Henri Poincaré.
   Google Scholar
• Christol, G. (1985). Diagonales de fractions rationnelles et équations de Picard-Fuchs. In Amice, Y., Christol, G., Robba, P., eds. Study group on ultrametric analysis, 12th year, 1984/85, No. 1, pages Exp. No. 13, 12. Paris: Secrétariat Math.
   Google Scholar
• Chyzak, F. (1997). An extension of Zeilberger’s fast algorithm to general holonomic functions. Vol. 217, pages 115–134. 2000. Formal power series and algebraic combinatorics (Vienna).
   Google Scholar
• Cooper, S. (2012). Sporadic sequences, modular forms and new series for 1/π. Ramanujan J. 29(1–3):163–183.
   Web of Science ®Google Scholar
• Cooper, S. (2017). Ramanujan’s Theta Functions. Cham: Springer.
   Google Scholar
• Coster, M. J. (1988). Supercongruences. PhD thesis, University of Leiden.
   Google Scholar
• Chan, H. H., Tanigawa, Y., Yang, Y., Zudilin, W. (2011). New analogues of Clausen’s identities arising from the theory of modular forms. Adv. Math. 228(2):1294–1314.
   Web of Science ®Google Scholar
• Chan, H. H., Zudilin, W. (2010). New representations for Apéry-like sequences. Mathematika 56(1):107–117. doi:10.1112/S0025579309000436
   Web of Science ®Google Scholar
• Delaygue, É. (2018). Arithmetic properties of Apéry-like numbers. Compos. Math. 154(2):249–274.
   Web of Science ®Google Scholar
• Deutsch, E., Sagan, B. E. (2006). Congruences for Catalan and Motzkin numbers and related sequences. J. Number Theory, 117(1):191–215. doi:10.1016/j.jnt.2005.06.005
   Web of Science ®Google Scholar
• Dwork, B. (1969). p-adic cycles. Inst. Hautes Études Sci. Publ. Math. (37):27–115. doi:10.1007/BF02684886
   Google Scholar
• Gessel, I. (1982). Some congruences for Apéry numbers. J. Number Theory 14(3):362–368. doi:10.1016/0022-314X(82)90071-3
   Web of Science ®Google Scholar
• Gorodetsky, O. (2019). q-congruences, with applications to supercongruences and the cyclic sieving phenomenon. Int. J. Number Theory 15(9):1919–1968.
   Web of Science ®Google Scholar
• Kazandzidis, G. S. (1969). On congruences in number-theory. Bull. Soc. Math. Grèce (N.S.), 10(fasc., fasc. 1):35–40.
   Google Scholar
• Kummer, E. E. (1852). Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math. 44:93–146.
   Google Scholar
• Lipshitz, L. (1988). The diagonal of a D-finite power series is D-finite. J. Algebra 113(2):373–378. doi:10.1016/0021-8693(88)90166-4
   Web of Science ®Google Scholar
• Lucas, E. (1988). Sur les congruences des nombres eulériens et les coefficients différentiels des functions trigonométriques suivant un module premier. Bull. Soc. Math. Fr. 6:49–54.
   Google Scholar
• MacMahon, P. A. (1960). Combinatory analysis. Two volumes (bound as one). New York: Chelsea Publishing Co.
   Google Scholar
• McIntosh, R. J. (1992). A generalization of a congruential property of Lucas. Amer. Math. Monthly 99(3):231–238. doi:10.1080/00029890.1992.11995840
   Web of Science ®Google Scholar
• Mimura, Y. (1983). Congruence properties of Apéry numbers. J. Number Theory 16(1):138–146. doi:10.1016/0022-314X(83)90038-0
   Web of Science ®Google Scholar
• Moy, R. (2013). Congruences among power series coefficients of modular forms. Int. J. Number Theory 9(6):1447–1474. doi:10.1142/S1793042113500371
   Web of Science ®Google Scholar
• Malik, A., Straub, A. (2016). Divisibility properties of sporadic Apéry-like numbers. Res. Number Theory 2:Art. 5, 26.
   Web of Science ®Google Scholar
• Mellit, A., Vlasenko, M. (2016). Dwork’s congruences for the constant terms of powers of a Laurent polynomial. Int. J. Number Theory 12(2):313–321. doi:10.1142/S1793042116500184
   Web of Science ®Google Scholar
• Osburn, R., Sahu, B. (2011). Congruences via modular forms. Proc. Amer. Math. Soc. 139(7):2375–2381.
   Web of Science ®Google Scholar
• Osburn, R., Sahu, B. (2011). Supercongruences for Apéry-like numbers. Adv. in Appl. Math. 47(3):631–638.
   Web of Science ®Google Scholar
• Osburn, R., Sahu, B. (2013). A supercongruence for generalized Domb numbers. Funct. Approx. Comment. Math. 48(part 1):29–36. doi:10.7169/facm/2013.48.1.3
   Google Scholar
• Osburn, R., Sahu, B., Straub, A. (2016). Supercongruences for sporadic sequences. Proc. Edinb. Math. Soc. (2), 59(2):503–518.
   Web of Science ®Google Scholar
• Peters, C., Stienstra, J. (1989). A pencil of K3-surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero. In Barth, W. P., Lange, H., eds. Arithmetic of Complex Manifolds (Erlangen, 1988), Vol. 1399 of Lecture Notes in Math. Berlin: Springer; pp. 110–127.
   Google Scholar
• Robert, A. M. (2000). A Course in p-adic Analysis, Vol. 198 of Graduate Texts in Mathematics. New York: Springer-Verlag.
   Google Scholar
• Stienstra, J., Beukers, F. (1985). On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271(2):269–304.
   Web of Science ®Google Scholar
• Schmidt, A. L. (1995). Legendre transforms and Apéry’s sequences. J. Austral. Math. Soc. Ser. A 58(3):358–375.
   Google Scholar
• Sloane, N. J. A. (1996). The on-line encyclopedia of integer sequences. Available at: https://oeis.org.
   Google Scholar
• Strehl, V. (1994). Binomial identities—combinatorial and algorithmic aspects. Discret. Math. 136(1–3):309–346. Trends in Discrete Mathematics. doi:10.1016/0012-365X(94)00118-3
   Web of Science ®Google Scholar
• Straub, A. (2014). Multivariate Apéry numbers and supercongruences of rational functions. Algebra Number Theory 8(8):1985–2007.
   Web of Science ®Google Scholar
• Samol, K., van Straten, D. (2015). Dwork congruences and reflexive polytopes. Ann. Math. Qué. 39(2):185–203.
   Web of Science ®Google Scholar
• Straub, A., Zudilin, W. (2015). Positivity of rational functions and their diagonals. J. Approx. Theory 195:57–69. doi:10.1016/j.jat.2014.05.012
   Web of Science ®Google Scholar
• Verrill, H. A. (1999). Some congruences related to modular forms. Max Planck Institute for Mathematics. Available at: http://www.mpim-bonn.mpg.de/preprints?title=%22Some+congruences+related+to+modular+forms%22
   Google Scholar
• Verrill, H. A. (2001). Picard-Fuchs equations of some families of elliptic curves. In Proceedings on Moonshine and related topics (Montréal, QC, 1999), Vol. 30 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc.; pp. 253–268.
   Google Scholar
• Verrill, H. A. (2010).Congruences related to modular forms. Int. J. Number Theory 6(6):1367–1390. doi:10.1142/S1793042110003587
   Web of Science ®Google Scholar
• Zagier, D. (2009). Integral solutions of Apéry-like recurrence equations. In Harnad, J., Winternitz, P., eds. Groups and Symmetries, Vol. 47 of CRM Proc. Lecture Notes. Providence, RI: Amer. Math. Soc.; pp. 3349–3663.
   Google Scholar
• Zagier, D. (2018). The arithmetic and topology of differential equations. In Mehrmann, V., Skutella, M., eds. European Congress of Mathematics. Zürich: Eur. Math. Soc.; pp. 717–776.
   Google Scholar
• Zeilberger, D. (1991). The method of creative telescoping. J. Symbolic Comput. 11(3):195–204. doi:10.1016/S0747-7171(08)80044-2
   Web of Science ®Google Scholar